Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

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2:05 minutes

Problem 4.R.91

Textbook Question

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (2x +1)² dx

Verified step by step guidance

Step 1: Recognize that the integral involves a polynomial expression squared, (2x + 1)². Expand this expression using the formula (a + b)² = a² + 2ab + b².

Step 2: Expand (2x + 1)² to get 4x² + 4x + 1. Rewrite the integral as ∫ (4x² + 4x + 1) dx.

Step 3: Break the integral into separate terms: ∫ 4x² dx + ∫ 4x dx + ∫ 1 dx. This allows you to integrate each term individually.

Step 4: Apply the power rule for integration to each term. For ∫ xⁿ dx, the result is (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration. Use this rule to integrate 4x², 4x, and 1.

Step 5: Combine the results of the individual integrals and include the constant of integration, C, to express the final indefinite integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Indefinite Integral

An indefinite integral represents a family of functions whose derivative is the integrand. It is denoted by the integral sign followed by the function and 'dx', indicating integration with respect to x. The result includes a constant of integration (C) since the derivative of a constant is zero, meaning multiple functions can yield the same derivative.

Polynomial Expansion

Polynomial expansion involves rewriting a polynomial expression in a simplified form, often using the binomial theorem. For example, expanding (2x + 1)² results in 4x² + 4x + 1. This step is crucial for integrating polynomials, as it allows for easier application of integration rules.

Power Rule of Integration

The power rule of integration states that the integral of x raised to the power n is (x^(n+1))/(n+1) + C, where n ≠ -1. This rule simplifies the process of finding indefinite integrals of polynomial functions, making it essential for solving integrals like ∫(2x + 1)² dx after expansion.